Monday, March 05, 2007

"How good a mathematician was Einstein? What is the source of the rumor he flunked math? How much math did he need to know to write, for example, his 1905 papers, and his general relativity papers? He claims he was lousy. How seriously should we take those claims?"

What truth there is to this long-standing myth is, well, relative. Einstein did intensely dislike the schooling of the Germany of his youth with its focus on rote memorization and its militaristic attitude. His grades did show his disengagement at times.

It is true that he failed something, but it wasn't math. His father and uncle had an electronics manufacturing business that went bust when Einstein was a junior in high school and when they relocated it to Italy, Albert was left behind in Germany to finish his schooling. Einstein acquired a doctor's note that let him out of high school and he went to Italy, surprising his family. When they demanded that he go back and finish high school, he mentioned his own plan to take the entrance exam for the ETH (the Swiss version of MIT) a year early, allowing him to go to university without a high school diploma -- hardly a move that would be undertaken by someone failing mathematics! He did fail that exam, but not the math section. As a result, he finished his high school work at a more liberal school in Switzerland and passed the entrance exam, enrolling at the ETH the next year.

He was not a model college student either. One of his professors, a mathematical physicist named Hermann Minkowski who would later go on to write the paper "Space and Time" which gave us our standard geometrical interpretation of Einstein's special theory of relativity. Minkowski was very fond of saying things like

"The mathematical education of the young physicist was not very solid, which I am in a good position to evaluate since he obtained it from me in Zurich some time ago."

Minkowski's best friend was also a mathematical genius, the great David Hilbert, who developed a version of the theory of general relativity at the same time Einstein did, but from a very different direction, one that was much more based on formal mathematics than on the physical intuitions that guided Einstein. Hilbert, playing off of Minkowski, said,

"Every school child in the streets of Goettingen knew more mathematics than Einstein, but he was the one who got it."

You have to understand the relationship between mathematicians and physicists. Physicists see math a s a tool and acquire competence with exactly as much as they think they will need to do their work whereas mathematicians love mathematics for its own sake and pursue aspects that seem to have no applicability. Physicists razz mathematicians for being wasting their time in imaginary worlds while mathematicians return fire by calling physicists mathematical neanderthals. That's what you are seeing here.

What generally happens, of course, is that physicists make progress and suddenly realize that to move further, they need additional, stronger mathematical tools and then have to go crawling back to the mathematicians to teach them the things that previously had no applicability. To work on the general theory of relativity, for example, Einstein had to seek out his friend Marcel Grossman to school him in tensor calculus, something he paid no attention to because it wasn't a tool he needed in his toolbox previously.

So was Einstein bad at math? There certainly are some physicists whose mathematical work is more elegant, but he was in no way mathematically disinclined. Unless you look at him through the cynical eyes of a mathematician...

Nick asks,

"What were Einstein's views about God? When he said that, "God does not play dice," what did he mean by that?"

Einstein did refer to God and religion in numerous places in his writings, and he generally has three separate notions he is invoking in various contexts.

The first is the standard Judeo-Christian picture of an anthropomorphic Creator. Einstein clearly and explicitly rejects the magical invisible man in the sky hypothesis. He argues that there is no reason to believe in a God separate from the world who has a mind and will. The notion of institutional religion is one he strongly distances himself from even though he describes himself as a cultural Jew.

The second way he uses religious language is ethically. Einstein buys into what philosophers call the fact/value distinction -- that there are is claims about how the world is and ought claims about how we should act and that these are wholly different things. To the first he gives the name "science" to the latter he uses the term "religion" although he is not necessarily talking about a set of absolute moral commands and duties and definitely not those which require acting in line with the desires of a supernatural agent -- since he denies that one exists.

The third (and this is the one relevant to Nick's quotation) is what he termed his "cosmic religion," a pantheistic approach to the universe that Einstein sees as similar to that of the heretical Jewish philosopher Baruch Spinoza. All things are interconnected and God is the whole of the universe which is governed by rational well-defined principles and the search for those principles is an act of the greatest righteousness.

In this picture in which the universe is a neat, well-ordered place is the notion of determinism. If the universe is in a given state and you know the rules that govern it, then you should be able to completely determine the future states of the universe. Quantum mechanics, to which Einstein himself made a major contribution in the early days, has in its central description of the workings of the world a point of uncertainty. Given the complete state of the universe at a given time and the equations for quantum mechanics, what you arrive at is not a description of how the world will be, but a set of possibilities of how the world might be and the likelihood that you will find it in each of those possible states. This was entirely unacceptable to Einstein. It led him to hold that quantum mechanics was right as far as it goes, but that it just doesn't go far enough. the theory is an incomplete description of the workings of this well-governed world. There had to be other hidden variables that needed to be figured into the equations and that when they were, the theory would become deterministic and not probabilistic. The world was orderly, not random. "God does not play dice with the universe" is Einstein-speak for there exists a perfect physics textbook that lays out the simple and all-governing rules by which the state of the universe develops. this, of course, is a statement underlying the scientific worldview of Einstein, not one that comes from it and as such was recognized by Einstein as an article of faith, as a matter of religion in some sense of the word.